Re: complex orthogonal group


Tobias Fritz wrote:

> Is SO(n,C) generated by reflections (at non-null vectors)? Any hints or
> references would be appreciated!

The reflections generate the orthogonal group for any nondegenerate
quadratic form over any field of characteristic not 2. (It's the
orthogonal group, not SO; the reflections have det = -1.) Here's the
standard proof.

Lemma 1. There exist non-null vectors.

Proof. Take any nonzero v. If (v,v) = 0, there is at least some w
with (v,w) not = 0 (that's nondegeneracy). Then (v+w, v+w) = 2(v,w) + (w,w)
and (v-w,v-w) = -2(v,w) + (w,w). These cannot both be zero.

Lemma 2. If (x,x) and (y,y) are equal and nonzero, there is a
reflection taking x either to y or to -y.

Proof. Here (x+y, x+y) = 2(x,x) + 2(x,y) and (x-y,x-y) = 2(x,x) - 2(x,y),
so at least one of x+y, x-y is non-null. It's trivial to compute that
(when defined) the reflection reversing x+y sends x to -y, while the
reflection reversing x-y sends x to y.

Lemma 3. In the setting of lemma 2, there is an isometry that is
either a reflection or a product of two reflections that sends x to y.

Proof. If we can only get -y in Lemma 2, we follow by the reflection
reversing y.

Proof of Theorem. We use induction on the dimension (the result
is obvious in dimension 1). Take any isometry, f. Take any
non-null p. Set q = f(p). Then (q,q) = (p,p), so by Lemma 3
we can follow f by one or two reflections to get an isometry g
that fixes p. As p is non-null, the hyperplane H orthogonal to
span(p) is a complement to that span. The restriction of the
quadratic form to H then must be nondegenerate, and g must
take H to itself. On H, our g can be expressed as a product of
reflections (by induction), and those all leave p fixed. Now
just go back and reverse the reflections getting g from f.


One reference for this is Bourbaki, Algebre 9, Sect. 6, No. 4.
Another is Jacobson, Basic Algebra I (second edition), p. 371-2.
Jacobson goes on to prove the more precise statement that in
dimension n you need at most n reflections. This can also be
found (e.g.) in O'Meara, Introduction to Quadratic Forms, 102-103.


William C. Waterhouse
Penn State

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